WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 378 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) SlicingProof [LOWER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 7 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 303 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 80 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 1001 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
        (22) BOUNDS(1, INF)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   FROM(z0) -> c
   FROM(z0) -> c1
   SEL(0, cons(z0, z1)) -> c2
   SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
   MINUS(z0, 0) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c6
   QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   ZWQUOT(z0, nil) -> c8
   ZWQUOT(nil, z0) -> c9
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
   ZWQUOT(z0, z1) -> c13
   ACTIVATE(n__from(z0)) -> c14(FROM(z0))
   ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
   ACTIVATE(z0) -> c16

The (relative) TRS S consists of the following rules:

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   minus(z0, 0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   zWquot(z0, nil) -> nil
   zWquot(nil, z0) -> nil
   zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
   zWquot(z0, z1) -> n__zWquot(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   FROM(z0) -> c
   FROM(z0) -> c1
   SEL(0, cons(z0, z1)) -> c2
   SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
   MINUS(z0, 0) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c6
   QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   ZWQUOT(z0, nil) -> c8
   ZWQUOT(nil, z0) -> c9
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
   ZWQUOT(z0, z1) -> c13
   ACTIVATE(n__from(z0)) -> c14(FROM(z0))
   ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
   ACTIVATE(z0) -> c16

The (relative) TRS S consists of the following rules:

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   minus(z0, 0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   zWquot(z0, nil) -> nil
   zWquot(nil, z0) -> nil
   zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
   zWquot(z0, z1) -> n__zWquot(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   FROM(z0) -> c
   FROM(z0) -> c1
   SEL(0', cons(z0, z1)) -> c2
   SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
   MINUS(z0, 0') -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   QUOT(0', s(z0)) -> c6
   QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   ZWQUOT(z0, nil) -> c8
   ZWQUOT(nil, z0) -> c9
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
   ZWQUOT(z0, z1) -> c13
   ACTIVATE(n__from(z0)) -> c14(FROM(z0))
   ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
   ACTIVATE(z0) -> c16

The (relative) TRS S consists of the following rules:

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   sel(0', cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   minus(z0, 0') -> 0'
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0', s(z0)) -> 0'
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   zWquot(z0, nil) -> nil
   zWquot(nil, z0) -> nil
   zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
   zWquot(z0, z1) -> n__zWquot(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
FROM/0

----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   FROM -> c
   FROM -> c1
   SEL(0', cons(z0, z1)) -> c2
   SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
   MINUS(z0, 0') -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   QUOT(0', s(z0)) -> c6
   QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   ZWQUOT(z0, nil) -> c8
   ZWQUOT(nil, z0) -> c9
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
   ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
   ZWQUOT(z0, z1) -> c13
   ACTIVATE(n__from(z0)) -> c14(FROM)
   ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
   ACTIVATE(z0) -> c16

The (relative) TRS S consists of the following rules:

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   sel(0', cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   minus(z0, 0') -> 0'
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0', s(z0)) -> 0'
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   zWquot(z0, nil) -> nil
   zWquot(nil, z0) -> nil
   zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
   zWquot(z0, z1) -> n__zWquot(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7

----------------------------------------

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SEL, activate, ACTIVATE, MINUS, QUOT, minus, sel, quot

They will be analysed ascendingly in the following order:
activate < SEL
ACTIVATE < SEL
activate < sel
quot < activate
QUOT < ACTIVATE
MINUS < QUOT
minus < QUOT
minus < quot

----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7


Generator Equations:
gen_c2:c39_17(0) <=> c2
gen_c2:c39_17(+(x, 1)) <=> c3(gen_c2:c39_17(x), c14(c))
gen_0':s10_17(0) <=> 0'
gen_0':s10_17(+(x, 1)) <=> s(gen_0':s10_17(x))
gen_cons:nil:n__from:n__zWquot11_17(0) <=> nil
gen_cons:nil:n__from:n__zWquot11_17(+(x, 1)) <=> cons(0', gen_cons:nil:n__from:n__zWquot11_17(x))
gen_c4:c512_17(0) <=> c4
gen_c4:c512_17(+(x, 1)) <=> c5(gen_c4:c512_17(x))
gen_c6:c713_17(0) <=> c6
gen_c6:c713_17(+(x, 1)) <=> c7(gen_c6:c713_17(x), c4)


The following defined symbols remain to be analysed:
MINUS, SEL, activate, ACTIVATE, QUOT, minus, sel, quot

They will be analysed ascendingly in the following order:
activate < SEL
ACTIVATE < SEL
activate < sel
quot < activate
QUOT < ACTIVATE
MINUS < QUOT
minus < QUOT
minus < quot

----------------------------------------

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s10_17(n15_17), gen_0':s10_17(n15_17)) -> gen_c4:c512_17(n15_17), rt in Omega(1 + n15_17)

Induction Base:
MINUS(gen_0':s10_17(0), gen_0':s10_17(0)) ->_R^Omega(1)
c4

Induction Step:
MINUS(gen_0':s10_17(+(n15_17, 1)), gen_0':s10_17(+(n15_17, 1))) ->_R^Omega(1)
c5(MINUS(gen_0':s10_17(n15_17), gen_0':s10_17(n15_17))) ->_IH
c5(gen_c4:c512_17(c16_17))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(12)
Complex Obligation (BEST)

----------------------------------------

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7


Generator Equations:
gen_c2:c39_17(0) <=> c2
gen_c2:c39_17(+(x, 1)) <=> c3(gen_c2:c39_17(x), c14(c))
gen_0':s10_17(0) <=> 0'
gen_0':s10_17(+(x, 1)) <=> s(gen_0':s10_17(x))
gen_cons:nil:n__from:n__zWquot11_17(0) <=> nil
gen_cons:nil:n__from:n__zWquot11_17(+(x, 1)) <=> cons(0', gen_cons:nil:n__from:n__zWquot11_17(x))
gen_c4:c512_17(0) <=> c4
gen_c4:c512_17(+(x, 1)) <=> c5(gen_c4:c512_17(x))
gen_c6:c713_17(0) <=> c6
gen_c6:c713_17(+(x, 1)) <=> c7(gen_c6:c713_17(x), c4)


The following defined symbols remain to be analysed:
MINUS, SEL, activate, ACTIVATE, QUOT, minus, sel, quot

They will be analysed ascendingly in the following order:
activate < SEL
ACTIVATE < SEL
activate < sel
quot < activate
QUOT < ACTIVATE
MINUS < QUOT
minus < QUOT
minus < quot

----------------------------------------

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7


Lemmas:
MINUS(gen_0':s10_17(n15_17), gen_0':s10_17(n15_17)) -> gen_c4:c512_17(n15_17), rt in Omega(1 + n15_17)


Generator Equations:
gen_c2:c39_17(0) <=> c2
gen_c2:c39_17(+(x, 1)) <=> c3(gen_c2:c39_17(x), c14(c))
gen_0':s10_17(0) <=> 0'
gen_0':s10_17(+(x, 1)) <=> s(gen_0':s10_17(x))
gen_cons:nil:n__from:n__zWquot11_17(0) <=> nil
gen_cons:nil:n__from:n__zWquot11_17(+(x, 1)) <=> cons(0', gen_cons:nil:n__from:n__zWquot11_17(x))
gen_c4:c512_17(0) <=> c4
gen_c4:c512_17(+(x, 1)) <=> c5(gen_c4:c512_17(x))
gen_c6:c713_17(0) <=> c6
gen_c6:c713_17(+(x, 1)) <=> c7(gen_c6:c713_17(x), c4)


The following defined symbols remain to be analysed:
minus, SEL, activate, ACTIVATE, QUOT, sel, quot

They will be analysed ascendingly in the following order:
activate < SEL
ACTIVATE < SEL
activate < sel
quot < activate
QUOT < ACTIVATE
minus < QUOT
minus < quot

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s10_17(n608_17), gen_0':s10_17(n608_17)) -> gen_0':s10_17(0), rt in Omega(0)

Induction Base:
minus(gen_0':s10_17(0), gen_0':s10_17(0)) ->_R^Omega(0)
0'

Induction Step:
minus(gen_0':s10_17(+(n608_17, 1)), gen_0':s10_17(+(n608_17, 1))) ->_R^Omega(0)
minus(gen_0':s10_17(n608_17), gen_0':s10_17(n608_17)) ->_IH
gen_0':s10_17(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7


Lemmas:
MINUS(gen_0':s10_17(n15_17), gen_0':s10_17(n15_17)) -> gen_c4:c512_17(n15_17), rt in Omega(1 + n15_17)
minus(gen_0':s10_17(n608_17), gen_0':s10_17(n608_17)) -> gen_0':s10_17(0), rt in Omega(0)


Generator Equations:
gen_c2:c39_17(0) <=> c2
gen_c2:c39_17(+(x, 1)) <=> c3(gen_c2:c39_17(x), c14(c))
gen_0':s10_17(0) <=> 0'
gen_0':s10_17(+(x, 1)) <=> s(gen_0':s10_17(x))
gen_cons:nil:n__from:n__zWquot11_17(0) <=> nil
gen_cons:nil:n__from:n__zWquot11_17(+(x, 1)) <=> cons(0', gen_cons:nil:n__from:n__zWquot11_17(x))
gen_c4:c512_17(0) <=> c4
gen_c4:c512_17(+(x, 1)) <=> c5(gen_c4:c512_17(x))
gen_c6:c713_17(0) <=> c6
gen_c6:c713_17(+(x, 1)) <=> c7(gen_c6:c713_17(x), c4)


The following defined symbols remain to be analysed:
QUOT, SEL, activate, ACTIVATE, sel, quot

They will be analysed ascendingly in the following order:
activate < SEL
ACTIVATE < SEL
activate < sel
quot < activate
QUOT < ACTIVATE

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SEL(gen_0':s10_17(n2171_17), gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17))) -> *14_17, rt in Omega(n2171_17)

Induction Base:
SEL(gen_0':s10_17(0), gen_cons:nil:n__from:n__zWquot11_17(+(1, 0)))

Induction Step:
SEL(gen_0':s10_17(+(n2171_17, 1)), gen_cons:nil:n__from:n__zWquot11_17(+(1, +(n2171_17, 1)))) ->_R^Omega(1)
c3(SEL(gen_0':s10_17(n2171_17), activate(gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17)))), ACTIVATE(gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17)))) ->_R^Omega(0)
c3(SEL(gen_0':s10_17(n2171_17), gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17))), ACTIVATE(gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17)))) ->_IH
c3(*14_17, ACTIVATE(gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17)))) ->_R^Omega(1)
c3(*14_17, c16)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
FROM -> c
FROM -> c1
SEL(0', cons(z0, z1)) -> c2
SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2))
MINUS(z0, 0') -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
QUOT(0', s(z0)) -> c6
QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
ZWQUOT(z0, nil) -> c8
ZWQUOT(nil, z0) -> c9
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1))
ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3))
ZWQUOT(z0, z1) -> c13
ACTIVATE(n__from(z0)) -> c14(FROM)
ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1))
ACTIVATE(z0) -> c16
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
minus(z0, 0') -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
zWquot(z0, nil) -> nil
zWquot(nil, z0) -> nil
zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))
zWquot(z0, z1) -> n__zWquot(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__zWquot(z0, z1)) -> zWquot(z0, z1)
activate(z0) -> z0

Types:
FROM :: c:c1
c :: c:c1
c1 :: c:c1
SEL :: 0':s -> cons:nil:n__from:n__zWquot -> c2:c3
0' :: 0':s
cons :: 0':s -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c2 :: c2:c3
s :: 0':s -> 0':s
c3 :: c2:c3 -> c14:c15:c16 -> c2:c3
activate :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
ACTIVATE :: cons:nil:n__from:n__zWquot -> c14:c15:c16
MINUS :: 0':s -> 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
QUOT :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c4:c5 -> c6:c7
minus :: 0':s -> 0':s -> 0':s
ZWQUOT :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> c8:c9:c10:c11:c12:c13
nil :: cons:nil:n__from:n__zWquot
c8 :: c8:c9:c10:c11:c12:c13
c9 :: c8:c9:c10:c11:c12:c13
c10 :: c6:c7 -> c8:c9:c10:c11:c12:c13
c11 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c12 :: c14:c15:c16 -> c8:c9:c10:c11:c12:c13
c13 :: c8:c9:c10:c11:c12:c13
n__from :: 0':s -> cons:nil:n__from:n__zWquot
c14 :: c:c1 -> c14:c15:c16
n__zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
c15 :: c8:c9:c10:c11:c12:c13 -> c14:c15:c16
c16 :: c14:c15:c16
from :: 0':s -> cons:nil:n__from:n__zWquot
sel :: 0':s -> cons:nil:n__from:n__zWquot -> 0':s
quot :: 0':s -> 0':s -> 0':s
zWquot :: cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot -> cons:nil:n__from:n__zWquot
hole_c:c11_17 :: c:c1
hole_c2:c32_17 :: c2:c3
hole_0':s3_17 :: 0':s
hole_cons:nil:n__from:n__zWquot4_17 :: cons:nil:n__from:n__zWquot
hole_c14:c15:c165_17 :: c14:c15:c16
hole_c4:c56_17 :: c4:c5
hole_c6:c77_17 :: c6:c7
hole_c8:c9:c10:c11:c12:c138_17 :: c8:c9:c10:c11:c12:c13
gen_c2:c39_17 :: Nat -> c2:c3
gen_0':s10_17 :: Nat -> 0':s
gen_cons:nil:n__from:n__zWquot11_17 :: Nat -> cons:nil:n__from:n__zWquot
gen_c4:c512_17 :: Nat -> c4:c5
gen_c6:c713_17 :: Nat -> c6:c7


Lemmas:
MINUS(gen_0':s10_17(n15_17), gen_0':s10_17(n15_17)) -> gen_c4:c512_17(n15_17), rt in Omega(1 + n15_17)
minus(gen_0':s10_17(n608_17), gen_0':s10_17(n608_17)) -> gen_0':s10_17(0), rt in Omega(0)
SEL(gen_0':s10_17(n2171_17), gen_cons:nil:n__from:n__zWquot11_17(+(1, n2171_17))) -> *14_17, rt in Omega(n2171_17)


Generator Equations:
gen_c2:c39_17(0) <=> c2
gen_c2:c39_17(+(x, 1)) <=> c3(gen_c2:c39_17(x), c14(c))
gen_0':s10_17(0) <=> 0'
gen_0':s10_17(+(x, 1)) <=> s(gen_0':s10_17(x))
gen_cons:nil:n__from:n__zWquot11_17(0) <=> nil
gen_cons:nil:n__from:n__zWquot11_17(+(x, 1)) <=> cons(0', gen_cons:nil:n__from:n__zWquot11_17(x))
gen_c4:c512_17(0) <=> c4
gen_c4:c512_17(+(x, 1)) <=> c5(gen_c4:c512_17(x))
gen_c6:c713_17(0) <=> c6
gen_c6:c713_17(+(x, 1)) <=> c7(gen_c6:c713_17(x), c4)


The following defined symbols remain to be analysed:
sel
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_0':s10_17(n6565_17), gen_cons:nil:n__from:n__zWquot11_17(+(1, n6565_17))) -> gen_0':s10_17(0), rt in Omega(0)

Induction Base:
sel(gen_0':s10_17(0), gen_cons:nil:n__from:n__zWquot11_17(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
sel(gen_0':s10_17(+(n6565_17, 1)), gen_cons:nil:n__from:n__zWquot11_17(+(1, +(n6565_17, 1)))) ->_R^Omega(0)
sel(gen_0':s10_17(n6565_17), activate(gen_cons:nil:n__from:n__zWquot11_17(+(1, n6565_17)))) ->_R^Omega(0)
sel(gen_0':s10_17(n6565_17), gen_cons:nil:n__from:n__zWquot11_17(+(1, n6565_17))) ->_IH
gen_0':s10_17(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
BOUNDS(1, INF)